Inverse Leidenfrost Effect: Levitating Drops on Liquid NitrogenM. Adda-Bedia,† S. Kumar,† F. Lechenault,† S. Moulinet,† M. Schillaci,† and D. Vella*,‡

†Laboratoire de Physique Statistique, Ecole Normale Superieure, Sorbonne Universites UPMC, CNRS, 24 rue Lhomond, 75005 Paris,France‡Mathematical Institute, Andrew Wiles Building, University of Oxford, Woodstock Rd, Oxford OX2 6GG, United Kingdom

*S Supporting Information

ABSTRACT: We explore the interaction between a liquid drop (initially atroom temperature) and a bath of liquid nitrogen. In this scenario, heattransfer occurs through film-boiling: a nitrogen vapor layer develops that maycause the drop to levitate at the bath surface. We report the phenomenologyof this inverse Leidenfrost effect, investigating the effect of the drop size anddensity by using an aqueous solution of a tungsten salt to vary the dropdensity. We find that (depending on its size and density) a drop eitherlevitates or instantaneously sinks into the bulk nitrogen. We begin bymeasuring the duration of the levitation as a function of the radius R anddensity ρd of the liquid drop. We find that the levitation time increasesroughly linearly with drop radius but depends weakly on the drop density. However, for sufficiently large drops, R ≥ Rc(ρd), thedrop sinks instantaneously; levitation does not occur. This sinking of a (relatively) hot droplet induces film-boiling, releasing astream of vapor bubbles for a well-defined length of time. We study the duration of this immersed-drop bubbling finding similarscalings (but with different prefactors) to the levitating drop case. With these observations, we study the physical factors limitingthe levitation and immersed-film-boiling times, proposing a simple model that explains the scalings observed for the duration ofthese phenomena, as well as the boundary of (R,ρd) parameter space that separates them.

■ INTRODUCTIONWhen a liquid droplet is deposited onto a hot solid surface thedrop may, provided the solid surface is hot enough, levitate on acushion of its own vapor. This phenomenon was described firstby Leidenfrost in the 18th century1 and is therefore known asthe Leidenfrost effect. Similarly, the critical substrate temper-ature above which the “film-boiling” that makes levitationpossible is called the Leidenfrost temperature. In everyday life,the Leidenfrost effect is used to test whether a frying pan is hotenough: flicking a drop of water into the pan will lead to thedrop skiting about only if the pan is above the Leidenfrosttemperature of water (around 200 °C). Although classical, thisintriguing effect has experienced a renaissance of research effortrecently with a number of studies focusing on the origin of thefilm-boiling regime2,3 and its relation to surface properties.4

Other studies have focused on the possibility of generatingmotion in this way;5−10 indeed the sublimation of dry ice(CO2) blocks, the solid analogue of the Leidenfrost effect, hasbeen proposed both as the cause of the eroded gullies that areobserved on Mars11 and a basic mechanism that could poweran engine there.12

In these common scenarios, levitation is limited by the timetaken for the levitating object to evaporate. An alternativescenario is the “inverse” Leidenfrost effect13 in which a waterdroplet is placed on the surface of a bath of a cryogenic liquidwith low boiling point, e.g., liquid nitrogen. In this case, thedeposited droplet causes the evaporation of the substrate,keeping its own mass constant during levitation. Instead ofbeing limited by the mass of the drop, the duration of the

inverse Leidenfrost effect is limited by the heat energy storedwithin the drop: levitation can only occur while the dropremains hot enough to sustain the film boiling of the liquidnitrogen. After levitation ceases, the drop comes into contactwith the bath, which (since the density of water is greater thanthat of liquid nitrogen) usually means that the drop then sinksbelow the surface,14−19 even though floating should be possiblefor sufficiently small drops.20 This phenomenon has beeninvestigated in relation to the film-boiling-mediated heattransfer between two liquids at different temperatures, whichoccurs in propellant spillage accidents:21 if liquid propellantspills accidentally during the test-stand or launching operationsof a rocket, a catastrophic explosion (resulting from thedetonation of the fuel and oxidizer) is possible.14,21

Even though the basic mechanism behind this liquid-on-liquid levitation is essentially the same as that responsible forthe usual Leidenfrost effect, our quantitative understanding of itremains only partial.14−18 Previous studies focused on modelingthe duration of levitation and its dependence on the radius ofthe drop.15,16,18 However, due to the relative paucity ofexperimental data, which in any case concentrated on simpleliquid drops,14,15,17 a complete understanding of the inverseLeidenfrost mechanism is lacking. Furthermore, levitation isonly possible for sufficiently small droplets; above a critical size,droplets immediately sink and undergo film-boiling within the

Received: February 16, 2016Revised: April 6, 2016

Article

pubs.acs.org/Langmuir

© XXXX American Chemical Society A DOI: 10.1021/acs.langmuir.6b00574Langmuir XXXX, XXX, XXX−XXX

bulk. Here, we explore this phenomenon using water basedsolutions at various densities and droplet radii. We are then ableto investigate the conditions that allow for the droplets tolevitate as well as the duration of the inverse Leidenfrost effectin each of the floating/levitating and sinking states.This paper is organized as follows: we begin by giving a more

detailed account of our experimental protocol, before givingsome raw experimental results. These experiments highlightsome discrepancies with previously proposed models and so wethen go on to develop a new mathematical model to describethe levitation and film-boiling regimes. This allows us toreassess the experimental data, by making a quantitativecomparison between theory and experiment.

■ EXPERIMENTAL METHODS AND OBSERVATIONSOur experiments were conducted in a homemade cryostat designed toallow imaging at the boiling temperature of nitrogen. The cryostat isprincipally built from a styrofoam box (labeled 1 in Figure 1). To have

a side view of the experiment, an opening was cut in one wall and twoplexiglass plates glued (one to each side), making a double-panedwindow (labeled 2 in Figure 1). To avoid the window frosting, acontinuous flow of dry air at room temperature was maintained withinthe gap. The box is closed with a plexiglass plate on its top, keeping asmall opening in order to allow gaseous nitrogen to escape.With the details so far described, this cryostat does not insulate the

liquid nitrogen well: liquid nitrogen placed inside a beaker boils,creating a rough, bubbling surface. This is not suitable for experimentson the inverse Leidenfrost effect, which require a relatively calm,smooth liquid surface to avoid premature sinking of droplets. Toremedy this problem, we used two different nitrogen baths: a sacrif icialbath (labeled 3 in Figure 1) contained within a large crystallizing dish(⌀ = 14 cm) and sitting on the bottom of the cryostat. The sacrificialbath boils permanently, producing a continuous flow of gaseousnitrogen at its boiling temperature. The cryostat is then completelyfilled with a thermalized atmosphere of gaseous nitrogen. Within thisbath we then place a study bath (labeled 4 in Figure 1), which iscontained in a beaker sitting on an aluminum block. As this bath issurrounded by boiling nitrogen and its vapor, it is well isolated(thermally) and does not spontaneously boil. Its surface is smooth andonly evaporates when heated by the proximity of a relatively warmdroplet. Within this simple experimental design, we are able toaccurately measure the levitation time of a droplet and capture itsdynamics (or to measure the film-boiling time for an immerseddroplet).

We deliver an aqueous droplet to the surface of the study bath usinga micropipette. The use of a micropipette allows for droplets ofeffective spherical radius, R = (3V/4π)1/3 (based on the dropletvolume V imposed with an accuracy of 2 μL), in the range 0.5 mm ≤ R≤ 3 mm. This droplet is initially at room temperature (and henceliquid) but cools and ultimately freezes, becoming opaque in theprocess (Figure 2a). In some cases, a droplet breaks into two parts

during freezing. We suspect that this behavior originates from theinitial formation of an outer frozen shell, which is broken as theremainder of the drop freezes and, hence, expands. However, the factthat this does not always occur indicates that such an outer shell doesnot always form, and hence that the heat transfer from the droplet tothe study bath is, in general, anisotropic.

If several droplets are deposited on the interface at the same time,they move together and (if this aggregation occurs before the dropsfreeze) coalesce into a larger drop. We do not study this aggregationhere, but attribute it to the interfacial deformations caused by eachdroplet, which are well-known to cause floating bodies to aggregatethe “Cheerios effect”.22 However, even an isolated droplet moveslaterally at the surface (see movie in the Supporting Information); webelieve that this motion is caused by small imbalances in the film ofvapor on which the droplet levitates. We do not study this motion herebut conjecture that a small perturbation in one direction leads to a self-

Figure 1. Schematic of the homemade cryostat used in ourexperiments. The cryostat is built from a 15 × 20 × 18 cm3 styrofoambox (1) in which a double-paned window is mounted (2) to allow aside-view of the experiment. To avoid condensation and frosting ofthis window, a flux of dry air is maintained between the panes. Asacrificial bath (3) sits on the bottom of the cryostat to preventvigorous boiling of the study bath (4), which is contained in a beakersitting on an aluminum block within the sacrificial bath.

Figure 2. (a) Closeup of a droplet (volume V = 70 μL, effective radiusR ≈ 2.5 mm, and density ρd = 1.04 g/cm3) levitating at the surface ofthe study bath. We see that the droplet remains approximatelyspherical while levitating. (b) The same drop after sinking is no longersurrounded by a vapor film and the emission of bubbles stops almostimmediately. (c) A drop of volume V = 70 μL, effective radius R ≈ 2.5mm, and density ρd = 1.08 g/cm3 that does not levitate initially insteadsinks to the bottom of the study bath and undergoes film boiling there.Here the drop sits at the bottom of the beaker and emits a string ofrising bubbles, which can be seen, albeit slightly blurred by their fastmotion, above the drop. (d) When the vapor film disappears, emissionof bubbles stops almost immediately revealing that the frozen drop hasadopted a “pear-shaped” form. In both (a) and (c) the drop issurrounded by a vapor film: its apparent thick dark surface is due tointernal total reflection on the interface between the liquid and gaseousnitrogen. (See also the movie in the Supporting Information.)

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perpetuating asymmetry and hence to sustained motion in thedirection of the perturbation, analogous to the “walking droplets”observed on a vibrating liquid bath.23

Depending on the size and density of the droplets, the initiallevitation reported above may be replaced by instantaneous sinking ofthe droplet. As well as varying the radius of the droplet (as describedabove), we therefore varied the droplet density, ρd. To achieve a broadrange of density variation, we used various dilutions of LST FastFloat(low toxicity sodium heteropolytungstates dissolved in water, suppliedby SIGMA); this allowed us to create drops with densities in the range1.02 g/cm3 ≤ ρd ≤ 2.8 g/cm3 (undiluted FastFloat). For a givendroplet density, levitation is possible only if the radius is below acritical value: above this critical value, the droplet sinks into the liquidbut remains surrounded by a film of vapor. The immersed drop thenprovokes vigorous boiling of the surrounding liquid nitrogen. Thedurations of levitation and this immersed-boiling are, for a given valueof drop radius, of the same order of magnitude. In the cases of bothlevitation and immersed boiling, the droplet ultimately freezes but thefinal frozen shape of the droplet in each case is quite different: dropsthat levitate remain roughly spherical (Figure 2b), while those that sinkwhile undergoing boiling exhibit a pear shape (Figure 2d).Examination of video footage (Supporting Information) shows thatthese drops are spherical when fully liquid (so that the shear stress ofthe rising bubbles does not significantly deform them). Instead, webelieve that the vapor layer is thinner at the base than the top, andhence that cooling occurs more rapidly at the base. In this way therising bubbles guide the expansion of the droplet on freezing to formthe tip of the “pear”. While this shape is reminiscent of the cusplikesingularity when a drop sitting on a plane is frozen from below,24 ourdrop has a slightly different form as the heat exchange occurs througha hemispherical surface and the vigorous film-boiling smooths thecusp.

■ EXPERIMENTAL RESULTS

Using the experimental apparatus described above, we haveinvestigated the inverse Leidenfrost behavior of droplets over alarge range of radii and densities. Two key questions that weseek to address are the following: (i) For how long does theinverse Leidenfrost phenomenon persist? (ii) For which valuesof radius R and density ρd does the droplet levitate at thesurface of the bath, rather than becoming immersed in thebath? We find that, for a given density, droplets levitateprovided that their radius is below some critical value, R <Rc(ρd); this is precisely analogous to the surface-tension-supported flotation of small dense objects, which sink above acritical size that depends on their density.20 Here we thereforeconsider first the question of the levitation time of drops forwhich R < Rc(ρd) before extending this to the case of immerseddroplets, R > Rc(ρd). Finally, we describe the boundary betweenlevitation and immersion, i.e., we study the form of Rc(ρd).Levitation Time. Our measurements of the time for which

a drop is able to levitate (see open circles in Figure 3) suggestthat this time scale is dependent on the drop radius but isrelatively insensitive to the density: with almost a factor of 3increase in density, we see an increase of only a few seconds inthe levitation time. However, as already indicated, themaximum radius for which levitation is possible does dependon the density.Immersed Boiling Time. Drops with a given density, ρd,

are only able to levitate if their radius (or equivalently volume)is below a critical value, R < Rc(ρd). Larger dropsinstantaneously become immersed in the liquid bath, sinkingto the bottom of the study bath. Nevertheless, such drops stillexhibit the Leidenfrost phenomenon: liquid nitrogen boils,rising from the drop in a column that destabilizes into bubbles.The measured bubbling time (the time for which the

Leidenfrost phenomenon is observed) is shown by the crossesin Figure 3. The bubbling time of immersed drops is shorterthan the levitation time of drops with similar radius(presumably due to the fact that they lose heat over a largersurface area). However, this bubbling time is of the same orderof magnitude as the levitation time, and depends on the dropradius in a similar manner.

Floating vs Sinking Condition. Finally, we investigate theregimes of (ρd,R) parameter space for which the dropletlevitates, or sinks. In the first case, the vapor produced by theLeidenfrost effect is enough to keep the droplet at the surface.In the second, the bubbles that the drop generates are notenough to lift it back to the surface and it remains immersed atthe bottom of the study bath.Figure 4 presents the experimentally determined regime

diagram, and shows the regions of (ρd,R)-space for which eachof these possibilities is observed. We observe that, for a givendroplet density, there is a sharp levitation-immersed boilingtransition at some critical radius, Rc(ρd). We further note thatRc(ρd) appears to be a decreasing, continuous function of ρd:even droplets with a very large density compared to that of theliquid nitrogen are able to levitate provided they are smallenough.

■ MODELING APPROACHESBased on our experiments, we make three observations of thissystem that need to be explained quantitatively: the levitationtime of droplets (i) varies approximately linearly with the dropradius and (ii) is approximately independent of the dropletdensity; finally, (iii) there is a sharp transition betweenlevitation and immersed bubbling. We first turn to see which,if any, of these observations may be explained by existingmodels.

Previous Modeling Approaches. The mechanismsinvolved in the inverse Leidenfrost effect have beed describedpreviously.14,15 The essential ingredient for levitation is that

Figure 3. Experimentally measured Leidenfrost time tL as a function ofradius of the drop R for various different drop densities. Each pointhere is the mean of at least 10 repeat experiments, and the error barsare smaller than the marker used. Experimental results are shown forboth drops levitating at the interface (○) and for those thatimmediately sink, becoming immersed in the nitrogen bath (×).The density of the droplet is encoded by its color with redrepresenting neat (undiluted) FastFloat (ρd = 2.8 g/cm3) and bluethe lightest droplets, which have ρd = 1.02 g/cm3. Note that the valuesof R and ρd correspond to those at room temperature (300 K), and inthe liquid state.

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heat passes from the (relatively warm) drop to the liquid bathbeneath it and, in so doing, causes that liquid to evaporate. Thisleads to the formation of a supporting vapor layer; in this layer,vapor is created and squeezed between the deformed interfaceand droplet surface eventually escaping into the nitrogenatmosphere within the cryostat. The pressure field required toperform this squeezing balances the weight of the droplet,allowing it to levitate at the interface. Of course, the dropletcontains only a finite amount of thermal energy and, since theevaporation of the liquid nitrogen requires the supply of latentheat to the bath, the temperature of the droplet decreases untilit first freezes itself (releasing latent heat of fusion in theprocess) before cooling further and ultimately reaches theLeidenfrost temperature of the liquid nitrogen. At this point,film boiling ceases, the vapor layer disappears and the dropletsinks into the bath.To quantify the above description requires a model of how

the mean vapor layer thickness, h, and the temperature of thedroplet, T, evolve. A key ingredient in this evolution is clearlythe heat transfer that occurs both within the droplet andthrough the vapor layer. Previous detailed studies14,15 of thefilm heat transfer coefficient found it to be k(1/R + 1/h) with kthe thermal conductivity of nitrogen vapor. It is clear that, for athin layer, h/R ≪ 1, the 1/R term should be neglected incomparison to the 1/h term (as noted originally14,15).However, subsequent works16,17 retained instead the 1/Rterm; a scaling analysis shows that this leads to a quadraticdependence of the floating time on the radius of the drop (seediscussion after eq 19 this paper), at odds with ourexperimental results, which suggest something closer to linear.Nevertheless, we note that the levitation times predicted by thismodel are of the correct order of magnitude according toexperiments.14−17 As well as this confusion over the appropriateheat transfer coefficient, previous studies solved the heattransfer problem (i.e., the evolution of the droplet temperature)under the assumption that the film thickness h does not varywith time. In reality, h evolves as the temperature differencethat drives evaporation decreases; in particular, as thetemperature difference driving nitrogen boiling decreases, the

film thickness must also decrease to ensure that sufficienthydrodynamic pressure is supplied for the droplet to float.Finally, and because these studies did not consider the verticalforce balance on the droplet, there is no previous criterion fordetermining whether a drop of a given density and radius willinitially levitate or become immersed.

Simplifications and Assumptions. The thermal con-ductivities of water, k ≈ 0.6 W/mK and ice k ≈ 2 W/mK, aresignificantly larger than that of nitrogen vapor, 2 × 10−2 W/mK≲ k ≲ 0.1 W/mK depending on the temperature.25 As a result,while the droplet remains liquid (so that convection mayredistribute heat), and even once frozen, we expect thetemperature profile within the droplet to remain approximatelyuniform and thermal conduction across the vapor layer to berate limiting; this corresponds to assuming that the Biotnumber (the ratio of conductive flux within the drop to that outof the drop) is small. In this small Biot number limit, therelevant length scale for conduction is only the (unknown)thickness of the thin Leidenfrost layer, h(θ, t). We furtherassume that the energy conducted across this gap isimmediately converted into Latent heat of vaporization of thegas, and so neglect the thermal profile within the liquidnitrogen bath. The gas layer around the drop is assumed to bethin so that the flow of the vapor beneath the drop may bedescribed by lubrication theory.26

In general, the thickness profile (i.e., the spatial variation) ofthe vapor layer on which the droplet levitates is not known. Forthe canonical problem of a droplet levitated above a rigidsubstrate (as in the usual Leidenfrost problem), this problemhas been solved in detail27−29 and shows that for levitatingdroplets larger than the capillary length a vapor-filled dimplemay form beneath the drop, as is observed experimentally.30,31

For drops smaller than the capillary length, the bottom of thedroplet approximately conforms to the flat substrate.29,31 Here,the vapor layer is bounded by two deformable interfaces so thatthe appropriate capillary length is larger than that of the bareinterface: the effective surface tension is larger and theappropriate density is the difference between the droplet andthe bath. Our droplets are therefore in the “small droplet”regime, so that we may simplify the problem (to allow as muchanalytical progress as possible) by assuming that the thicknessof the vapor layer is approximately uniform, i.e., h(θ, t) = h(t)(i.e., that the drop and bath interfaces conform to one another,as in the small droplet limit of the regular Leidenfrostproblem). Furthermore, the surface tension of the drop−vapor interface, γdv ≈ 72.8 mN/m, is significantly larger thanthat of the bath−vapor interface, γN ≈ 8.85 mN/m). It istherefore energetically favorable for the droplet interface to bedeformed as little as possible, at the expense of the bathinterface, and so we assume that the shape of the levitatingdroplet is spherical, in agreement with our experimentalobservations in Figure 2a and previous observations of dropletslevitating on a vibrated bath (but without phase change).32 Thesimplified problem is illustrated schematically in Figure 5.Finally, the physical parameters of the system (e.g., the

thermal conductivity k and vapor density ρg) vary withtemperature. We neglect these variations here, again to facilitateanalytical progress. The values of the various parameters thatwe use here are given in Appendix A; generally we use valuesfor the vapor at a temperature of 200 K, which is roughlymidway between the temperature of the study bath (T = 77.36K) and room temperature. (Choosing a round number for the

Figure 4. At a fixed droplet density, ρd, only drops below some criticaleffective radius, Rc(ρd), will perform the Leidenfrost phenomenonwhile levitating at the interface. Above this maximum radius, dropletssink into the liquid, becoming fully immersed, and perform theLeidenfrost phenomenon while immersed. Here points show theexperimentally determined value of Rc(ρd) for droplets of differentdensity, ρd, and volumes, V, with the effective radius defined to be R =(3V/4π)1/3. All points carry error bars, though these are many timessmaller than the actual size of the point on the graph.

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temperature has the advantage that the thermal properties ofnitrogen gas have been tabulated at intervals of 100 K.25)Conservation Laws and Solution. Our model for the

levitation of a droplet through the inverse Leidenfrostphenomenon is based on the conservation of thermal energy,the conservation of mass (of vapor) and vertical force balance(on the drop). Since it is the temperature difference betweenthe droplet and the bath, ΔT = Td − TN, that drives the inverseLeidenfrost effect, we begin by using conservation of energy todiscover how this “excess temperature” evolves. The situation iscomplicated by the fact that the droplet solidifies as it levitates.There are therefore three stages of levitation: in stage I thedroplet is liquid and the driving temperature ΔT evolves; instage II the droplet freezes, releasing the latent heat of fusionbut maintaining a constant ΔT (assuming conduction of thelatent heat is sufficiently fast) and in stage III the droplet issolid and the driving temperature ΔT evolves once again.In stage I, the global conservation of energy requires that the

rate at which the thermal energy of the (liquid) drop, ρdVcpΔT,decreases, must equal the rate at which energy diffuses acrossthe Leidenfrost layer, kΔT/h, integrated over the surface area,A, of the drop that is in close proximity to the nitrogen bath.We therefore have that

ρΔ = − Δ

tT

kAc V

Th

dd

( )d p (1)

This is an ordinary differential equation for ΔT but involves theunknown thickness of the Leidenfrost layer, h(t); an additionalequation is required to determine h(t), as we shall discussshortly. A similar equation holds for the evolution of the excesstemperature in stage III once the drop has solidified; however,the product of the liquid density and heat capacity, ρdcp, isreplaced by that of the solid droplet, ρd

(s)cp(s).

In stage II, the freezing of the droplet, it is the latent heat offusion f that must be diffused away over the liquid gap. If wedenote by ϕl the proportion of drop that is in the liquid state(so that ϕl = 1 at the beginning of stage II), then theconservation of energy here reads

ϕ

ρ= − Δ

tkAV

Th

d

dl

df

(2)

In each stage, therefore, we have an expression for the evolutionof either the driving temperature difference ΔT (stages I andIII, eq 1) or the liquid fraction within the droplet (stage II, eq2). However, each of these evolution equations depends on the(currently unknown) gap thickness h(t). We therefore nowturn to determining the evolution of h(t).We assume that the thickness h(t) is determined by the

requirement that the vapor (which is generated by the heat fluxfrom the droplet) be able to drain out of the layer. To proceed,we need to determine the areal flux (a volume per unit time,per unit surface area) q of nitrogen vapor that is generated byphase change. This flux q is determined by calculating thevolume q whose latent heat of vaporization, ρ qg

v , would carry

away the instantaneous heat flux, kΔT/h, across the layer, i.e.,ρ= Δq k T h/( )g

v . Since the gap between the droplet and the

bath is assumed thin, we then use lubrication theory to describethe motion of the nitrogen vapor (of viscosity μ) within thislayer and to determine the pressure field, p(θ, t), that drives thisflow. Combining the standard local conservation of massequation in lubrication theory26 with the source term thatcomes from the generation of vapor, we have that

θ θθ

μ θ ρ− ∂

∂∂∂

= Δ⎛⎝⎜

⎞⎠⎟

ht R

hR

p k Th

dd

1sin

sin12 g

v

3

(3)

Equation 3 may be solved, subject to a condition that p remainsfinite at the base of the drop (θ = 0) and that the pressure isatmospheric at some θ = θm, i.e. p(θ = θm) = 0. The pressurewithin the vapor layer is then

μρ

θθ

= − − Δ⎛⎝⎜⎜

⎞⎠⎟⎟

⎡⎣⎢

⎤⎦⎥p

Rh

ht

k Th

24 dd

logcos( /2)

cos( /2)

2

3g

vm (4)

Note that the maximum angular position θm also enters theconservation of energy eq 1 or 2 through the area A over whichconduction happens.To progress further, we need to determine from eq 4 an

equation for dh/dt. This is achieved in different ways for eachof the two cases (levitating drops and immersed drops), and soin Appendix B we consider each of these cases separately.However, the result of each calculation is that the evolutionequation for the gap thickness may be written in the form

ρ− Δ = −h

tk T

hch

dd g

v3

(5)

for some dimensional constant c that varies with drop size,density, etc.To make further progress, we introduce dimensionless

variables

ρρ ρ* = Δ * = Δ * =h hA V T

cT t

kAc V

t/ , ,d p

gv

2

d p2

(6)

Here, the gap thickness is essentially nondimensionalized bythe radius of the sphere, R, temperatures by the temperaturechange of the droplet brought about by changing a unit mass ofliquid nitrogen into gas (modified by the density ratio betweenthe gas and the drop), and time is nondimensionalized by thetime scale of thermal conduction over the radius of the drop. Inthese variables, the evolution eq 1 for stage I of the processbecomes

Figure 5. Model setup considered in the theoretical calculationspresented here: a spherical droplet sits at the interface of the liquidnitrogen bath on a cushion of nitrogen vapor that makes up theLeidenfrost layer. Here we assume that the thickness of this layer isindependent of angular position θ, but dependent on time, i.e., h(θ, t)= h(t).

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*Δ * = −

Δ **t

TT

hd

d( )

(7)

while for stage II (freezing), we have the dimensionless versionof eq 2

ϕ ρ

ρ*= −

Δ **tT

hd

dl g

v

df

(8)

instead of eq 7. In stage III, eq 7 is modified to read

*Δ * = −

Δ **t

Tc

c

Th

dd

( ) p

p(s)

(9)

In each stage, the gap thickness evolves according to thedimensionless version of eq 5, which may be written

** =

Δ **

− *th

Th

Chd

d3

(10)

where the dimensionless constant

ρπμ

ρ

ρ ρ θ

μ θ

=−

⎧

⎨⎪⎪

⎩⎪⎪

C

gV

R A

c

k

gV c

k RA

6, floating

(1 cos )

24 log[1/cos( /2)], immersedN

d5

4 4d p

4d p m

4m (11)

where ρN is the density of liquid nitrogen. Using parametervalues typical of our experiments, we find that C ≫ 1 (typically105 ≲ C ≲ 108). In this regime, eq 10 ensures that the thicknessof the film evolves approximately quasi-statically, i.e., h∗ ≈(ΔT∗/C)

1/4 (see Appendix C). The largeness of C ensures thath ≪ 1, i.e., that the cushioning layer of vapor is only a smallfraction of the drop radius, which is a required condition forour use of lubrication theory to be valid.In this quasi-static limit, we may eliminate h∗ ≈ (ΔT∗/C)

1/4

from the appropriate form of conservation of energy and solvethe resulting equation. For example, from eq 7, we find that

Δ * = * − *TC

t t4

( )4c 4

(12)

and

* =Δ * = * − *

⎛⎝⎜

⎞⎠⎟h

TC

t t14

( )1/4

c

(13)

where the constant of integration

* = Δ *t T C4[ (0)/ ]c 1/4(14)

is determined by the initial condition. Physically, the constant t∗c

corresponds to the (dimensionless) time at which the excessheat of the droplet would vanish if it remained liquidthroughout (i.e., in stage I). Of course, the duration of theLeidenfrost phenomenon in this case is more complicated, notleast because of the three stages through which the motionpasses. We therefore turn now to consider the total duration ofthe Leidenfrost phenomenon.Duration of the Leidenfrost Phenomenon. The

duration of the first two stages of the motion are relativelysimple to calculate since, by definition, stage I ends once theexcess temperature ΔT reaches a critical value ΔTF = Tfreeze −TN (with Tfreeze = −10 °C the estimated freezing temperature ofthe polytungstate solution). In dimensionless terms,

* = Δ * − Δ *tC

T T4

[ (0) ](I)1/4

1/4F,

1/4

(15)

Stage II lasts as long as is required to dissipate the latent heat offusion. During this stage, the drop remains at Tfreeze. Thedimensionless duration reads

ρρ* = Δ *

−tC

T1(II)1/4

df

gv F,

3/4

(16)

The final phase of motion is stage III in which the droplet hasfrozen but continues to cool to the temperature of the bath. Weshall assume that this phase ceases when the temperature of thedroplet reaches the Leidenfrost temperature of nitrogen, Tc ≈126 K.14 The dimensionless duration of stage III is thus

* = Δ − * − **tC

c

cT T T

4[ ( ) ](III)

1/4p(s)

pF,

1/4L N

1/4

(17)

and the total dimensionless duration of the Leidenfrostphenomenon is

* = * + * + * ∝ −t t t t CL (I) (II) (III) 1/4(18)

The experimentally measured Leidenfrost time (nondimension-alized as in eq 6), t*

L, is plotted as a function of thedimensionless parameter C in Figure 6. This shows two key

features of the data: (i) experimental results over a range of sizeand density (and floating in different states) approximatelycollapse onto a single curve and (ii) the theoretical predictionin eq 18) gives a reasonable estimate of the observedLeidenfrost time (recalling that there are no fitting parametersin our model), though the model does overestimate theduration of the Leidenfrost phenomenon. Furthermore, theexperiments suggest a power law behavior t*

L ∼ C−1/4, as

Figure 6. Dimensionless levitation time t∗L plotted as a function of the

single dimensionless parameter C given by eq 11. As in Figure 3,experimental results are shown for both drops levitating at theinterface (○) and for those which immediately become immersed inthe nitrogen (×). The density of the droplet is encoded by its colorwith red representing neat FastFloat (ρ = 2.8 g/cm3) and blue thelightest droplets (very dilute solutions of FastFloat) for which ρ = 1.02g/cm3; a more detailed color map is given in the legend. The solid lineshows the expression 18 obtained from this theory with a finaltemperature at the end of the Leidenfrost phenomenon of TL = 126K.16 Note that, in evaluating the dimensionless constant C and thetheoretical predictions, we have taken the values of k and μ at 200 K,as detailed in Appendix A.

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F

predicted in eq 18. Finally, given that the duration of each stageof the process scales with C in the same way means that we canestimate the proportions that the drop spends in each stage; wefind that the ratios t(I):t(II):t(III) ≈ 1:2:3each stage occupies asignificant fraction of the whole experiment.In dimensional terms, eq 18 may be written

ρ ρρ

= −

− − + −

− − + − −

⎛⎝⎜⎜

⎞⎠⎟⎟t

C

c V

kA

cT T

T Tc

cT T

T Tc

T T

4{( )

( ) [( )

( ) ] ( ) }

L 1/4d p

2

2d p

gv

1/4

0 N1/4

freeze N1/4 p

(s)

pfreeze N

1/4

L N1/4

f

pfreeze N

3/4

(19)

Since V/A ∼ R and C ∼ R3, we see immediately that tL ∼ R5/4,which is almost linear in the droplet radius. Similarly, sincealmost all occurrences of the drop density ρd are incombination with the corresponding specific heat capacity cpand if we assume, as discussed in Appendix A, that ρdcp ≈ cstfor our solutions, then eq 19 suggests that the Leidenfrost timeis almost independent of the droplet density (with at most aρd

1/4 dependence, observed in the levitating regime).Scaling Analysis of Leidenfrost Duration. The detailed

theory above shows that the levitation time scales approx-imately linearly with drop radius, in contrast with mostpreceding analyses, which suggested a quadratic scaling.Furthermore, this time is approximately independent of thedrop density. To understand these two surprising observationsbetter, it is worth re-evaluating our model with a brief scalinganalysis of the problem. Our key modeling assumption is thatthe relevant heat flux limiting the cooling of the droplet (andhence the duration of the Leidenfrost phenomenon) is theconduction of heat across the vapor layer, of thickness h. Wetherefore have, in scaling terms, that both the heat flux and thevolume flux of vapor into the lubricating layer, q ∼ ΔT/h. Todetermine the gap thickness h we used lubrication theory,which showed that the pressure drop within the lubricatinglayer Δp ∼ μuR2/h2 (see eq 3) with u a typical horizontalvelocity scale, which must, by mass conservation, be u ∼ q/h.We therefore have Δp ∼ μqR2/h3 ∼ μΔTR2/h4. This pressuredrop must be equated to either the pressure drop in theexternal liquid (in the immersed case) or the additionalpressure required to support the drop (in the levitating case);either way, we have on purely dimensional grounds that Δp ∼ρ∗gR (with the value of the appropriate density ρ∗ dependingon the situation) so that ρ∗gR ∼ μΔTR2/h4 and hence h ∼(μΔTR/ρ∗g)1/4. Crucially, this gap thickness is only weaklydependent on the size of the droplet, R, and the droplet densityρd. The surface-integrated heat flux out of the droplet, ∼ΔTR2/h, and so the time to lose the initial heat energy of the droplet ist ∼ Rh/ΔT ∼ R5/4/ρ∗

1/4, which is approximately linear in theradius R and independent of the droplet density (since ρdcp isapproximately constant for dilute salt solutions, as discussed inAppendix A). This argument makes it clear that the real root ofthe quasi-linear dependence of the Leidenfrost time on dropletradius we observe is that thermal conduction happens over thethin cushioning layer, of thickness h that depends only weaklyon R.Transition from Floating to Sinking. We have now

understood the duration of the Leidenfrost phenomenon.

However, we have not yet discussed whether a drop of a givendensity and volume will initially levitate at the interface or willimmediately sink, becoming immersed in the liquid andundergoing film-boiling. In this section, we seek to understandthis question by studying the largest droplet that is able to floatfor a liquid of a given density. To do this, we draw on theconsiderable body of work on the “floatability” of small, denseobjects, which has seen intense interest in recent years.20

As already assumed in our thermodynamic model, therelatively large tension of the vapor−water interface (relative tothat of the liquid−vapor) will ensure that the droplet remainsrelatively spherical. We therefore treat the Leidenfrost dropletas a perfectly nonwetting (contact angle = 180°) sphere andcalculate the maximum density a nonwetting sphere of givenradius can have for equilibrium flotation to remain possible.This calculation follows closely that presented elsewhere20,33

and so here we simply present our numerical results. In thisscenario, it is the size of the sphere relative to the capillarylength of the interface between the liquid nitrogen and gaseousnitrogen, lc = (γN/ρNg)

1/2, that is important: spheressignificantly smaller than the capillary length can be supportedat the interface even if their density is larger than that of thesupporting liquid. In particular, the maximum floatable radius ata given density is

ρρ

≈⎛⎝⎜⎜

⎞⎠⎟⎟R l

3

2cN

d

1/2

(20)

for R ≪ lc.Figure 7 shows the data from Figure 4 replotted in

dimensionless terms and compared with the results of this

calculation. We see that the experiments do not violate thetheoretical upper bound for nonwetting spheres floating at aliquid interface: levitating Leidenfrost drops are no better atfloating at the interface than would be a nonwetting sphere ofthe same density. Indeed we see that drops significantly belowthe maximum floatable density actually sink, becomingimmersed in the liquid nitrogen bath. One possibility for thisdiscrepancy is that the drops are (by necessity) dropped from a

Figure 7. Maximum radius for which levitation is possible, Rc, as afunction of the density ratio, ρd/ρN, for a perfectly nonwetting sphereis shown by the solid blue curve. The experimentally determinedmaximum radius for drops of FastFloat at different density are shownby the points and lie below the theoretical maximum. The discrepancymight be attributed to the fact that impact induces prematuresinking.20

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G

height onto the bath. It has been shown previously that impactat a finite speed may cause the premature sinking of solidspheres34 and cylinders.35,36 Indeed, even if an object is placedat the interface with zero velocity, the acceleration due togravity in reaching its equilibrium floating depth may inducesinking.35

■ DISCUSSION AND CONCLUSIONS

In this paper, we have presented a series of experiments on theinverse Leidenfrost effect for a liquid drop on a bath of liquidnitrogen; these experiments revealed that the levitation timescales approximately linearly with drop radius, in contrast withmost preceding analyses, which suggested a quadratic scaling.Furthermore, this time is approximately independent of thedrop density. We were able to explain these two experimentalobservations using a theory that coupled heat conduction to theforce balance condition required as a result of levitation; theessential ingredients of this theory may be understood from thescaling analysis given above.While the density of the droplet has little effect on the

duration of the Leidenfrost phenomenon, it does play a majorrole in the phenomenology since, for a given radius, the densityof the drop determines whether it floats or sinks. Ourexperiments suggest that this transition should be understoodwithin the context of the vast literature on the floating/sinkingtransition of dense interfacial objects, rather than being aparticular feature of the Leidenfrost phenomenon.Looking to the future, we hope that other groups will be

motivated to extend our theoretical model to study the effect ofspatial variations in the gap thickness h, which we have assumedis uniform (but varying in time). This has been done in detailfor the regular Leidenfrost problem29,37 and for dropletslevitating on a vibrating bath in the absence of phase change.32

It may also be interesting to investigate the thermal propertiesof the polytungstate solutions used hereis our assumption ofa constant volumetric specific heat ρdcp and latent heat ρd

v

responsible for the discrepancy between experiment and theoryobserved here? Other intriguing phenomena worth studyinginclude the self-propulsion of the drops studied here and that ofother drops (e.g., ethanol droplets) that do not sink, even afterthe Leidenfrost phenomenon has ceased (presumably due tosurface tension effects and the smooth surface of frozenethanol). This long-term continuation of levitation (or, moreaccurately, floating), may result in a frozen droplet repeatedlybouncing off the edge of the container (see SupportingInformation movie). Finally, we note that more violent versionsof the Leidenfrost effect are believed to occur when alkalimetals are placed at the surface of water and may ultimately bethe cause of the explosions that are usually attributed to thehydrogen that is produced via chemical reactions.38

■ APPENDIX A: PARAMETER VALUES

In performing our calculations we have used a range ofparameter values from the literature. We summarize these inTable 1 for ease of reference and to aid navigation of the manyrelevant papers.For the thermodynamic properties of the polytungstate

solutions, we emphasize that the solutions used are prepared atrelatively low concentration despite the dramatic increase indensity: the concentration of neat FastFloat is approximately0.6 mol/L, i.e., a molar fraction of 10−2. We therefore expectthat the molal heat capacity of the solution remains dominated

by that of the solvent, water. By neglecting the volume changedue to dilution, we adopt the simplifying assumption of aconstant volumetric heat capacity: ρdcp ≈ cst. (For morecommon salts, such as NaCl and KCl, ρdcp varies by less than5% for concentrations up to 1 mol/L.39) For similar reasons,we assume that ρd

f is independent of the concentration.

■ APPENDIX B: DETAILS OF PRESSURE CLOSUREIn this Appendix, we discuss the remaining condition needed toturn the pressure profile (eq 4) into an equation for theevolution of the gap thickness, h(t). There are two cases toconsider: the cases of a levitating and of an immersed droplet.

Floating/Levitating. For a Leidenfrost droplet to levitate atthe interface, the vertical component of the pressure force(integrated over the surface of the droplet) must balance theweight of the droplet, i.e.,

∫ρ π θ θ θ θ=θ

gV R p t2 ( , )cos sin dd2

0

m

(21)

Substituting for p(θ, t) from eq 4 into the above expression andeliminating ΔT/h in favor of dΔT/dt using eq 1, we find that

ρρ

+ Δ = −⎛⎝⎜⎜

⎞⎠⎟⎟t

hc V

AT ch

dd

d p

gv

3

(22)

where

ρπμ θ θ

=− +

cgV

R3 (3 4cos cos 2 )d

4m m (23)

In this case, we recover the general form presented in eq 5.Immersed. In the immersed case, there is an unknown

reaction force from the base of the tank that balances theweight of the droplet. Instead of the global vertical forcebalance, therefore, we require that the change in pressure fromthe bottom of the droplet to the datum of pressure, taken at θ =θm, should balance the hydrostatic pressure change within theliquid over the same region, ρNgR(1 − cos θm). We thereforehave that

ρ θ

μρ θ

−

= − − Δ⎛⎝⎜⎜

⎞⎠⎟⎟

⎡⎣⎢

⎤⎦⎥

gR

Rh

hk T

h

(1 cos )

24log

1cos( /2)v

N m

2

3g m (24)

which can be rewritten in the form of eq 5 with constant

Table 1. Physical Parameter Values Used in the Model

quantity symbol value ref

Leidenfrost temperature TL 126 K 16viscosity @ 200 K μ 12.89 μPa·s 40thermal cond. @ 200 K k 18.72 mW/m/K 40density of gaseous N2 @ 200 K ρg 1.706 kg/m3 41

Latent heat of N2 vaporizationv 2 × 105 J/kg 42

density of liquid N2 ρN 807 kg/m3 43liquid droplet heat capacity ρdcp 4.2 MJ/K/m3 44

latent heat of H2O fusion f 334 kJ/kg 45

solid droplet heat capacity ρd(s)cp

(s) 1.9 MJ/K/m3 45N2 liquid−vapor tension γN 8.85 mN/m 46

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H

ρ θμ θ

=−

cg

R(1 cos )

24 log[1/cos( /2)]N m

m (25)

The Choice of θm. In both floating and immersed cases, wetake θm = π/2. This value is chosen for simplicity and torepresent a reasonable value based on experimental observa-tions: in the immersed case, a thin film is only observed aroundthe lower hemisphere since bubbling becomes violent in theupper hemisphere. Nevertheless, we emphasize that the relativeinsensitivity of the dimensionless constant C to the parameterθm means that the precise value of θm used should not be tooimportant.

■ APPENDIX C: NONLINEAR DYNAMICS OF ODEsIn this Appendix, we briefly discuss the general behaviour of theordinary differential eqs 7−10. The key observation is that thedimensionless parameter C is typically large (our experimentsgenerally have 105 ≲ C ≲ 108). In this limit, the evolutionrapidly converges onto the slow manifold h∗ ≈ (ΔT∗/C)

1/4 sothat the evolution is then quasi-static. This can be illustrated byplotting the trajectories (from the numerical solution of eqs 7and 10 for stage I of the phenomenon with a range of initialconditions in (ΔT∗, h∗) space), as shown in Figure 8.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.lang-muir.6b00574.

Video showing levitating, sinking, and self-propellingdrops (AVI)

■ AUTHOR INFORMATIONCorresponding Author*E-mail: dominic.vella@maths.ox.ac.uk.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

We thank Dr K. E. Daniels for her support and contributions tothis project. This project has received funding from theEuropean Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation program(Grant Agreement No. 637334 − GADGET).

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Figure 8. Numerical solution of the full system of ODEs for stage I ofthe Leidenfrost phenomenon, i.e., eqs 7 and 10 represented in(ΔT∗,h∗) phase space. Each solid curve represents a trajectory forgiven initial conditions with arrows showing the direction of increasingtime. The trajectories are initially very close to vertical showing thatthe system evolves approximately isothermally until it reaches thequasi-steady state h∗ = (ΔT∗/C)

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